reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem
  c in f"SD iff [c,f/.c] in f & f/.c in SD
proof
  thus c in f"SD implies [c,f/.c] in f & f/.c in SD
  proof
    assume
A1: c in f"SD;
    then
A2: (f qua Function).c in SD by GRFUNC_1:26;
A3: [c,(f qua Function).c] in f by A1,GRFUNC_1:26;
    then c in dom f by FUNCT_1:1;
    hence thesis by A3,A2,PARTFUN1:def 6;
  end;
  assume that
A4: [c,f/.c] in f and
A5: f/.c in SD;
  c in dom f by A4,Th46;
  hence thesis by A5,Th26;
end;
